Connection between the conductivity and bulk modulus of isotropic composite materials

Rigorous cross-property bounds that connect the effective electrical conductivity sigma* and the effective bulk modulus kappa* of any isotropic two-phase composite are derived when the volume fractions of the phases are either specified or unknown. These bounds enclose lens-shaped regions in the sigma*-kappa* plane, portions of which are attainable by certain microgeometries and thus are optimal. Our cross-property bounds apply also to anisotropic composites with cubic symmetry. The bounds are applied to some general situations, as well as to specific microgeometries, including regular and random arrays of spheres and hierarchical geometries corresponding to effective-medium theories. It is shown that knowledge of the effective conductivity can yield sharp estimates of the effective bulk modulus (and vice versa), even in cases where there is a wide disparity in the phase properties.